Optimal. Leaf size=112 \[ -\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g}-\frac {f p \text {Li}_2\left (-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{2 g^2}-\frac {p x^2}{2 g} \]
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Rubi [A] time = 0.19, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2475, 43, 2416, 2389, 2295, 2394, 2393, 2391} \[ -\frac {f p \text {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g}-\frac {p x^2}{2 g} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{g}-\frac {f \log \left (c (d+e x)^p\right )}{g (f+g x)}\right ) \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g}-\frac {f \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 g}\\ &=-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {\operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e g}+\frac {(e f p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 g^2}\\ &=-\frac {p x^2}{2 g}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}+\frac {(f p) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 g^2}\\ &=-\frac {p x^2}{2 g}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}-\frac {f p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 0.81 \[ -\frac {-\log \left (c \left (d+e x^2\right )^p\right ) \left (-e f \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+d g+e g x^2\right )+e f p \text {Li}_2\left (\frac {g \left (e x^2+d\right )}{d g-e f}\right )+e g p x^2}{2 e g^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.77, size = 672, normalized size = 6.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 123, normalized size = 1.10 \[ -\frac {{\left (\log \left (e x^{2} + d\right ) \log \left (\frac {e g x^{2} + d g}{e f - d g} + 1\right ) + {\rm Li}_2\left (-\frac {e g x^{2} + d g}{e f - d g}\right )\right )} f p}{2 \, g^{2}} - \frac {f \log \left (g x^{2} + f\right ) \log \relax (c)}{2 \, g^{2}} - \frac {{\left (e p - e \log \relax (c)\right )} x^{2} - {\left (e p x^{2} + d p\right )} \log \left (e x^{2} + d\right )}{2 \, e g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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